Lemma: Propagated ambiguity and its sign

For changes (eq:Dq), the change of the attached coefficient is

The root pairing is distributional, with a cutoff equal to one near its compact support if needed. There is no additional sign in (eq:deltaCK).

Proof. Lemma lem:attachK applies to any distribution in the relative variable, including ${\partial }^{\gamma }\delta$. Thus every changed product exists and has the same compact support bound.

Use the coordinates $(y,x,z)$, $z=y-w$, whose Jacobian has modulus one. The root and relative contact distribution become the tensor product $B_{f,g}(y,x)\otimes {{\partial }_z}^{\gamma }\delta (z)$. Pairing with a smooth coefficient gives

At fixed $y,x$, ${\partial }_z=-{\partial }_w$, so

The signs cancel.

Choose a cutoff in $(y,x)$ equal to one near the root support and a cutoff in $w$ equal to one near $\mathrm{supp}{\chi }_{\mathrm{I}}$. Since $a_{\mathsf{k}}$ contains the switch, multiplication by the latter cutoff leaves $a_{\mathsf{k}}$, and all its derivatives, unchanged. This justifies the displayed smooth pairing and proves (eq:deltaCK).